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Theorem dfpprod2 31964
Description: Expanded definition of parallel product. (Contributed by Scott Fenton, 3-May-2014.)
Assertion
Ref Expression
dfpprod2 pprod(𝐴, 𝐵) = (((1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V)))))

Proof of Theorem dfpprod2
StepHypRef Expression
1 df-pprod 31936 . 2 pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))
2 df-txp 31935 . 2 ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) = (((1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V)))))
31, 2eqtri 2642 1 pprod(𝐴, 𝐵) = (((1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V)))))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1481  Vcvv 3195  cin 3566   × cxp 5102  ccnv 5103  cres 5106  ccom 5108  1st c1st 7151  2nd c2nd 7152  ctxp 31911  pprodcpprod 31912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703  df-cleq 2613  df-txp 31935  df-pprod 31936
This theorem is referenced by:  pprodcnveq  31965
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